A sample approximation solution procedure for chance-constrained districting problems

In this paper, a Districting Problem with chance-constraints balancing requirements is investigated. The goal is to partition a set of basic Territorial Units into p contiguous and compact districts such that their probability of being balanced is above a minimum threshold. For such a problem, an approximate counterpart is considered, in which deterministic inequalities are used to express the need for the districting plan to be balanced across a large set of randomly drawn scenarios. This leads to a sample approximation problem. In order to solve the latter, a new heuristic algorithm is devised. The proposed procedure exploits a location–allocation scheme coupled with a so-called “balancing constraints-generation” procedure. In practice, the sample approximation problem is iteratively solved by adding demand scenarios (and, hence, the corresponding balancing constraints) on the fly. Several measures to drive the selection of such scenarios and embed them into the problem during the solution process are introduced and discussed. Extensive computational experiments on testbed instances from the literature prove the validity of the devised procedure, showing that it outperforms existing heuristics in the number of solved instances and/or computing times while assuring comparable solutions’ quality, especially for larger-sized test cases.